Some Results on the Intermediate Logics

Abstract

In ClOj, we developed the method of Kripke models and gave some applications of it to the study of the intermediate logics. We found that the use of Kripke models is very efficient, since in many cases the algebraic structure of Kripke models reflects well the properties of the logics characterized by them. In dll]], we proved that a certain relation holds between the logics characterized by some Kripke models and the logics having the finite model property. As we stated in the correction at the end of CllH, the original proof contained an error. So, we emphasize here that the following problem remains open: Has any intermediate logic a characteristic Kripke model? In this paper, we will proceed in the same direction as Hi OH anc* Ell]. At present, we have at hand many particular intermediate logics. But we have very little knowledge about the general properties common to many logics. For instance, though many logics having the disjunction property have been known, we don't know what conditions make a logic have the disjunction property. We think that the central aim of the study of intermedate logics is to construct the theory about the general properties of them. The notion of the slice introduced by Hosoi Q4] gave us the first clue to our purpose. We will introduce in §1 other classifications of intermediate logics. In §2, we will characterize them by Kripke models, just as the slice was characterized by the height of Kripke models in [110]. In §3, we will investigate about the disjunction property in connection with these classifications. We assume familiarity with the terminologies and the notions of ^10]. In ^10], we consider a